# What are some face and vertex transitive polyhedra that are not edge transitive?

A way I know to define the Platonic solids is that they are the only convex polyhedra that are edge, face, and vertex transitive.

If we retain only the vertex transitivity, one finds a new family of solids, the 13 (14?) Archimedean solids + the infinite series of prisms and antiprisms.

Similarly, retaining only face transitivity, one finds the 13 Catalan solids and the infinite series of bipyramids and trapezohedra. These solids are the duals of the vertex transitive solids.

2 Catalan solids and 2 Archimedean solids are also edge transitive.

But the definition of the Platonic solids as edge, face and vertex transitive seems to imply that there are face+vertex transitive solids that are not edge transitive. Something in between Catalan and Archimedean. If that is not the case, face and vertex transitivity would suffice to define the Platonic solids.

Which are these face & vertex transitive solids? (excluding the Platonic solids)

I know of only one example (which I suspect is the only convex example, though I have no proof for this), and that is a non-regular tetrahedron constructed from 4 congruent acute angled triangles. Since the edges have different lengths, it cannot possibly be edge-transitive. The faces are congruent, so it's face-transitive, and the verticies all share the same vertex figure, so it is also vertex-transitive.

There may or may not be non-convex examples, but I don't know anything about them.

• Interesting, any pictures? Commented Feb 5, 2022 at 13:45
• I was just about to add this one Commented Feb 5, 2022 at 13:47
• You got what I was looking for, I will leave the question open for a bit to see if we can find more. Two more questions come to mind, why is this not in the Catalan/Archimedean list? And is it sef-dual? Commented Feb 5, 2022 at 14:00
• It's not in those lists because it isn't a Catalan or Archimedean solid. It's not archimedean because its faces aren't regular, and it's not Catalan becuase it's not dual to an Archimedean solid. I do believe that the dual of any polyhedron in this family will also be in the family though. (and that would make sense with the tetrahedron being self-dual). Commented Feb 5, 2022 at 14:05

These are called noble polyhedra. The disphenoid tetrahedra mentioned by AshSeifert are indeed the only non-regular convex examples. This was proved by M Brückner in 1906.

Going beyond the convex polyhedra, the crown polyhedra are also noble.

An interesting document about them, including some of Brückner's figures, can be found at Exploring Noble Polyhedra With Stella4D